Levinson’s theorem for the Schrödinger equation in two dimensions

Levinson’s theorem for the Schrödinger equation with a cylindrically symmetric potential in two dimensions is reestablished by the Sturm-Liouville theorem. The critical case, where the Schrödinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison to Levinson’s theorem in the noncritical case, the half bound state for the [Formula Presented] wave, in which the wave function for the zero-energy solution does not decay fast enough at infinity to be square integrable, will cause the phase shift of the [Formula Presented] wave at zero energy to increase an additional [Formula Presented].